Properties

Label 2-570-19.9-c1-0-6
Degree $2$
Conductor $570$
Sign $0.980 - 0.196i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (0.766 + 0.642i)6-s + (1.32 − 2.29i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (−0.560 − 0.970i)11-s + (−0.499 + 0.866i)12-s + (2.61 + 2.19i)13-s + (2.49 + 0.907i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (−0.733 − 4.16i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.442 − 0.371i)3-s + (−0.469 + 0.171i)4-s + (0.420 + 0.152i)5-s + (0.312 + 0.262i)6-s + (0.501 − 0.868i)7-s + (−0.176 − 0.306i)8-s + (0.0578 − 0.328i)9-s + (−0.0549 + 0.311i)10-s + (−0.168 − 0.292i)11-s + (−0.144 + 0.249i)12-s + (0.724 + 0.608i)13-s + (0.666 + 0.242i)14-s + (0.242 − 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.178 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94656 + 0.193046i\)
\(L(\frac12)\) \(\approx\) \(1.94656 + 0.193046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-3.79 - 2.15i)T \)
good7 \( 1 + (-1.32 + 2.29i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.560 + 0.970i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.61 - 2.19i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.733 + 4.16i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-3.49 + 1.27i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.624 + 3.54i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.35 - 2.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.33T + 37T^{2} \)
41 \( 1 + (5.00 - 4.19i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-10.6 - 3.88i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.11 - 6.31i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.68 - 0.614i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.0320 + 0.181i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.54 + 2.01i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.16 + 12.3i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (8.89 + 3.23i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (5.10 - 4.27i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (4.73 - 3.97i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.37 - 9.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.03 + 7.58i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.21 - 6.89i)T + (-91.1 + 33.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.75788173238965682377838091493, −9.666947728167859738431264223699, −8.894875099969877924869056727398, −7.932666483382938365958930397457, −7.19854592979568717341569822069, −6.41328737729368404539528086520, −5.28273285184406116766726755047, −4.20144233494968550037507391814, −2.99932161110703699024559590419, −1.26788868023567438236746411225, 1.60340675070204504926089373819, 2.75544648098091635607753378770, 3.85098607040474288844192970026, 5.13997123247020908841861275462, 5.70775124689053019189277027109, 7.24718589481074621086746719238, 8.617742865894505678204975236135, 8.804695479959793098458644342323, 9.952010392329298559330782768476, 10.66004084462743996366412585467

Graph of the $Z$-function along the critical line